Ascent Trajectory Optimisation for a

Single-Stage-to-Orbit Vehicle with Hybrid Propulsion

Fabrizio Pescetelli∗, Edmondo Minisci†, Christie Maddock‡, Ian Taylor†,Richard E Brown§

Centre for Future Air-Space Transportation Technology, Dept. of Mechanical & Aerospace Engineering

University of Strathclyde, Glasgow G4 0LT, Scotland, UK

This paper addresses the design of ascent trajectories for a hybrid-engine, high per-formance, unmanned, single-stage-to-orbit vehicle for payload deployment into low Earthorbit. A hybrid optimisation technique that couples a population-based, stochastic algo-rithm with a deterministic, gradient-based technique is used to maximize the final vehiclemass in low Earth orbit after accounting for operational constraints on the dynamic pres-sure, Mach number and maximum axial and normal accelerations. The control search spaceis first explored by the population-based algorithm, which uses a single shooting methodto evaluate the performance of candidate solutions. The resultant optimal control law andcorresponding trajectory are then further refined by a direct collocation method basedon finite elements in time. Two distinct operational phases, one using an air-breathingpropulsion mode and the second using rocket propulsion, are considered. The presence ofuncertainties in the atmospheric and vehicle aerodynamic models are considered in orderto quantify their effect on the performance of the vehicle. Firstly, the deterministic optimalcontrol law is re-integrated after introducing uncertainties into the models. The proximityof the final solutions to the target states are analysed statistically. A second analysis isthen performed, aimed at determining the best performance of the vehicle when these un-certainties are included directly in the optimisation. The statistical analysis of the resultsobtained are summarized by an expectancy curve which represents the probable vehicleperformance as a function of the uncertain system parameters. This analysis can be usedduring the preliminary phase of design to yield valuable insights into the robustness of theperformance of the vehicle to uncertainties in the specification of its parameters.

Nomenclature

a Acceleration, m/s2 or speed of sound, m/sA Area, m2

c Specific heat capacity, J/K·kgc Control design vectorC Optimisation constraint

or aerodynamic coefficientD Drag, NEAS Equivalent air speed, m/sfa Fuel-air ratioF Force or thrust, Ng Gravitation acceleration, m/s2

h Altitude above mean sea level, m or kmIsp Specific impulse, slb Lower bound percentage, %

L Lift, Nm Mass, kgM Mach numberp Probability, %P Pressure, Paq Dynamic pressure, PaQr Heating value of the fuel, J/kgR Radius, mSE Sampling surfaceT Temperature, Kub Upper bound percentage, %v Velocity, m/sα Angle of attack, radγ Flight path angle, rad

∗Postgraduate Research student†Lecturer‡Research Fellow, AIAA Member§Professor, AIAA Senior Member

1 of 18

American Institute of Aeronautics and Astronautics

γi Isentropic indexδT Throttle controlε Angle between the thrust vector and

velocity vector of the vehicle, radε Uncertainty distribution functionη Efficiency, %θ Longitude, rad or deg E

λ Latitude, rad or deg Nµ Bank angle, radρ Density, kg/m3

χ Path direction angle, radω Angular velocity, rad/s

Subscript

0 Initial time, usually t = 0aero Aerodynamic modelatm Atmospheric modelc Cut-off valuecomb CombustionD Drage Exit of the engineE Earthf Final timehs Hypersonic

L Liftnom NominalP, Pg Propellant, Propellant post-combustionref References Statess SupersonicT Thrustunc Uncertaintywing Lifting surfaces

I. Introduction

Spurred by the retirement of the NASA Space Shuttle and the obsolescence of the current generation ofconventional launchers, the technology that will be required for the next generation of space-access vehicles iscurrently a very active field of research. A range of re-usable single-stage-to-orbit (SSTO) aerospace vehiclesis currently being considered, and hybrid engines that combine the high thrust of rocket engines with thelow fuel consumption of air-breathing engines at lower altitudes and Mach numbers are serious contendersfor their propulsion. The next generation of vehicles will be used to carry either payload or crews into space,and by emphasizing full re-usability in their design and employing an airline-like approach, where the costof acquisition is amortized over repeated flights, these vehicles promise to dramatically reduce the cost perkilogram of access to space. An example of such a vehicle is the Skylon space plane that is currently underdevelopment within the UK by Reaction Engines Ltd.1,2

The overarching aim of the research described in this paper is to develop a model-based software toolthat will aid in the preliminary design of the next generation of space-access vehicles. Within this paper, anascent trajectory, starting in the low supersonic regime and extending through to hypersonic velocities, isdesigned for a representative space plane concept by optimizing a control law that alters the angle of attackof the vehicle and the throttle setting of its engines to vary the flight time along its trajectory into orbit. Thekey performance index for a space-access vehicle is the maximum payload mass it can transport into space.The total mass fraction (i.e., fully loaded vehicle mass/initial mass) is limited principally by the achievablespecific impulse (Isp) of the engines.3

In the process of designing the next generation of air-space transportation systems, it is fundamental thatany new tools and approaches that are developed for the evaluation of vehicle performance are also capableof functioning reliably in an integrated design environment. As such, the optimal trajectories that resultfrom the application of these tools need to be robust, in the sense that cognizance must be taken of the effectof uncertainties within the various parameters and models that comprise the overall vehicle system design.During the process whereby the control law is optimized in order to maximize the total payload mass intoorbit (or equivalently to minimize the fuel consumption), it is thus also highly desirable to be able to assessthe sensitivity of the design to variations in, for instance, the performance of the engines (available thrust andmass flow rate), the vehicle aerodynamics (lift and drag coefficients) and, for a vehicle propelled by hybridengines, the effect of changing the speed and altitude at which its engines switch between air-breathing androcket propulsion modes.

In this vein, in addition to finding an optimal control law for the ascent trajectory of a representativeair-space transport vehicle, the impact on the predicted overall performance of the level of uncertaintyinherent in various parameters in the atmospheric and aerodynamic models used to characterize the vehicle

2 of 18

American Institute of Aeronautics and Astronautics

is also evaluated. While the analysis presented here is by no means comprehensive, the idea is to conducta preliminary assessment of the performance of the vehicle in the presence of uncertainties in its modelingand characterization.

Two different approaches are used. In the first approach, a set of trajectories are integrated usinguncertainties introduced into the system models but with the nominal optimal control law applied to thesystem. The second approach instead re-optimizes the control law, in the presence of uncertainties todetermine the effect on the vehicle performance, in this case using the final vehicle mass into orbit as thefigure of merit. A statistical analysis is conducted on the results in order to produce an expectancy curvewhich relates the probability of the vehicle having a certain performance to the expected variation due touncertainty in the system design parameters.

The foundation of an ideal robust and reliable method for trajectory optimisation is an algorithm whichguarantees convergence, with a high level of confidence, to the global optimum of the system. The convergenceof traditional gradient-based optimal control solvers is severely compromised by the presence of discontinuitiesin any of the design models, and it can be difficult, if not impossible, to find a valid optimal solution usingthis approach if a good initial guess is not properly chosen. One of the interesting elements of the workpresented here is that a hybrid stochastic-deterministic approach to the optimization of the system has beenused in order to overcome the limitations of traditional gradient-based techniques.

The paper starts by describing the methods and tools that were used to solve the trajectory optimisationproblem for the representative space plane and to characterize the various uncertainties in the models thatwere used in its design. In the subsequent section of the paper, the composition of the various models thatcomprise the overall system design for the space plane are described in more detail. The specific trajectoryoptimisation problem is then addressed, followed by a discussion of the results.

II. Optimisation and robustness analysis

The following section provides background into the different types of optimisation approaches that areused in this work to determine the optimal control law for the ascent trajectories of the space access vehiclesand explains the rationale behind the approach that has been adopted.

A. Trajectory optimisation

All practical methods for solving optimal control problems involves an approach which discretizes the systeminto a finite set of unknowns. Generally, the dynamical system, characterized by a set of continuous functions,is transcribed into a problem with a finite set of variables, following which the resultant finite dimensionalproblem is solved by using a numerical parameter optimisation method. A further step is required to assessthe accuracy of the finite dimensional approximation to the continuous problem. This methodology allowsthe optimal control problem to be re-written as a non-linear programming problem (NLP).

There are several methods that can be used to transcribe an optimal control problem into a NLP.4 Theeasiest approach is the direct shooting method, where only the system controls are discretized and the wholeoptimisation process iterates over the integration of dynamical system from initial to final conditions. Thedisadvantage of the approach is that the objective function and the constraints on the optimization canonly be evaluated at the end of the simulation. Transcription by the single shooting approach results in aNLP problem with a relatively small number of independent design variables, but dynamical systems whichare subject to instability for certain values of their control parameters (as is often the case) are difficult totreat with this method. A more sophisticated approach is the direct collocation method where both thecontrols and the state variables defining the system are discretized in terms of the elapsed time. In thisway, an infinite-dimensional ODE system is replaced by a finite number of equality constraints, and theintegrals associated with the objective and constraint functions of the original problem are approximated.This approach gives a large but sparse NLP, which can be solved conveniently using any one of a number ofsequential quadratic programming (SQP) solvers.5

The proposed trajectory optimization approach is based on a mixed formulation which combines apopulation-based stochastic algorithm with a deterministic gradient-based method. Population-based stochas-tic algorithms are able to explore the global search space very efficiently, and are thus able to find feasiblesolutions when the constraints on the system are sufficiently loose and the dimensionality of the problemis not high (generally less than 100 design variables). Sampling of the search space can become inefficient,

3 of 18

American Institute of Aeronautics and Astronautics

however, if the constraints on the system are overly strict. On the other hand, deterministic gradient basedsolvers can deal efficiently with equality constraints and problems with many dimensions (generally manymore than 100 design variables). The idea here is to use a stochastic approach coupled to a single shootingtranscription method to find an initial solution, in terms of the variation of the system states and controlswith time, for the optimized trajectory by first relaxing the constraints on the final states of the system. Thenthis solution is used as a starting point for a further local optimisation by the direct collocation method.This second step is intended to both improve the value of the objective function and to ensure that theequality constraints on the required final states are satisfied.

Figure 1: General scheme of the trajectory optimisation process.

A general overview of the optimisation process that was used in the work described in this paper is shownin Fig. 1 and is described in more detail as follows.

1. Transcribe the optimal control problem into a single shooting NLP As a first step, the opti-mal control problem is converted to a problem with a finite set of variables by using a direct singleshooting approach and relaxing the constraints on the system.

2. Solve the single shooting NLP using MOPED The NLP is then solved using a hybrid Evolution-ary Algorithm (EA) which is obtained by coupling together two different algorithms: the Multi-Objective Parzen-based Estimation of Distribution6 (MOPED) and a modified version of the Infla-tionary Differential Evolution Algorithm7,8 (IDEA).

The MOPED algorithm belongs to a subset of Evolutionary Algorithms called Estimation of Distribu-tion Algorithms9 (EDA). Within these algorithms the typical evolutionary search operators, such ascrossover and mutation, are replaced with statistical tools. These tools build an approximate proba-bilistic model of the search space, with the role of the crossover and mutation operators replaced byappropriate sampling of this model. MOPED is an EDA that can be used for multi-objective optimi-sation of continuous problems, and uses the Parzen method10 to build a probabilistic representation ofPareto solutions and can handle multivariate dependencies of the variables.6,11 This EDA optimizerimplements the general layout and selection techniques of the Non-dominated Sorting Genetic Algo-rithm II (NSGA-II),12 but with the traditional crossover and mutation search approaches of NSGA-IIreplaced by sampling of the Parzen model. NSGA-II was chosen as the basis for MOPED mainly dueto its simplicity and for the quality of the results that are obtained using this approach for variousdiverse optimisation problems. The Parzen method uses a non-parametric approach to kernel densityestimation. It allocates Nind identical elementary probability density functions (PDF) (where Nind isthe number of individuals within the current population), each one centered on a different element of

4 of 18

American Institute of Aeronautics and Astronautics

the sample. A probabilistic model of the promising search space portion is built on the basis of thestatistical data provided by Nind individuals, and new individuals are sampled by the probabilisticmodel itself. The variance of each elementary PDF depends on i) the location of the individuals inthe search space and ii) the fitness value of these individuals. The way the algorithm is constructedleads to values that favor sampling in the neighborhood of the most promising solutions. The featuresof MOPED, which is used here for a single objective problem, allows for an efficient exploration of thesearch space but often prevent fine convergence on the optimal point particularly when the solutions ofthe final population are spread over different areas which are far apart from each other in the feasiblespace. This feature of the approach has prompted the idea of coupling MOPED with another EAwhich has better convergence properties. To this end, IDEA has been selected.

3. Refine MOPED solution using IDEA The Inflationary Differential Evolution Algorithm (IDEA) isbased on a hybridization of a differential evolution13 (DE) variant and the logic behind monotonicbasin hopping14 (MBH). The resulting algorithm has been demonstrated to out-perform both DE andMBH on some difficult space trajectory design problems, for instance those where the search space hasa (multi) funnel-like structure.7 The main features of the IDEA algorithm are reported in depth byVasile et al.;7 here just the more practical aspects of the currently adopted algorithm are discussed. Inessence, the final solutions obtained by MOPED are clustered based on the Euclidean distance betweenthem in the search space, resulting in a variable number of solution clusters. The distance thresholdfor each cluster is chosen such that each cluster should have a number of individuals in the range[Ncl,l, Ncl,u]. A DE process is then performed a number of times, beginning with the sub-populationof each cluster. Each process is stopped only when the population contracts to below a predefinedthreshold. Every time the DE process stops, a local search is performed in order to properly convergeto the local optimum. As the design optimisation in this case is constrained, the internal DE mechanismcan be modified such that the comparison of individuals during the DE process is able to account for theconstraints on the system. In the unconstrained DE algorithm,13 each parent solution is compared withits offspring; the solution with the best objective function value is chosen to be part of the population ofthe next generation. Instead, for the constrained DE algorithm used here, when parents and offspringare compared, the solutions are first evaluated in terms of constraint compatibility. Evaluating thesolutions based first on how well they comply with the constraints ensures that each solution is validbefore optimality in terms of the objective function for the system is assessed.

4. Transcribe the optimal control problem using DFET The optimal control problem is convertedto a problem with a finite set of variables using a direct collocation method based on Finite Elementsin Time (DFET) on a spectral basis.15

5. Initialize DFET-based NLP using best solution from IDEA and solve using gradient methodThe NLP problem is then solved by a gradient-based optimisation method, which uses the solutionobtained from the previous stochastic optimisation as a starting point.

B. Robust design and uncertainties

Within the early stages of design, where models and simulations are used to assess preliminary design options,it is important to characterize the uncertainty affecting the models and to consider their impact on the designin order to properly assess the performance of the system. Uncertainties occur across all of the differentphases of modeling and simulation, and can be categorized either as aleatory uncertainties, which are dueto the inherent random behavior of a system, or epistemic uncertainties that arise from a lack of knowledgeabout a system.

Once the levels of uncertainty within the system have been estimated and implemented, they can be usedto i) estimate the sensitivity of the control law(s) to the uncertainties themselves (robustness analysis), ii)obtain a statistical characterization of the likely vehicle performance in the presence of uncertainty, and/oriii) obtain a robust optimization of the trajectory, in other words to optimize the performance of the systemwhile minimizing the effect of uncertainties. In this paper the estimated uncertainties have been used toanalyze the robustness of the nominal control law for the vehicle and to obtain a statistical characterizationof the vehicle performance in the presence of uncertainty in some of its key design variables.

Two types of uncertainty-based analyses are performed. The first test is aimed at evaluating the ro-bustness of the nominal control law against the estimated uncertainties. Once the optimisation has been

5 of 18

American Institute of Aeronautics and Astronautics

performed using nominal models, the nominal control law is reintegrated using atmospheric and aerodynamicmodels whose output variables have been perturbed relative to their nominal values. The second test insteadis aimed at evaluating the optimal performance of the vehicle subject to uncertainties within the optimisationloop. Multiple optimizations are performed, each one with a different randomized set of perturbations inthe atmospheric and aerodynamic models.

The level of uncertainty on a specific variable is represented by a percent deviation around the nominalvalue. Specifically, the atmospheric temperature and pressure are perturbed as any uncertainty consideredfor these two quantities affects other atmospheric values, such as the density and speed of sound, as well asother models such as the aerodynamics and propulsion which lead to altering the trajectory dynamics andultimately the control. Uncertainties within the aerodynamics model specifically are evaluated by perturbingthe net lift and drag forces.

Once a deterministic solution is available from the atmospheric and aerodynamic models, uncertaintiesare introduced by perturbing the nominal values of the outputs of the models. The value of a genericuncertain quantity, xnom, in the presence of uncertainty, xunc, can be defined as:

xunc = xnom + εSExnom (1)

where ε is the uncertainty bounding function which depends on the operational conditions, and SE is asampling surface mapping the set of operating conditions into the interval [−1, 1]. The function SE isdefined as an interpolating surface dependant on the value of each interpolating node. Each time a newperturbation profile is needed, a new set of values for the parameters defining SE is randomly generated. Forthe robustness analysis of the nominal control law, the process generates n different SE surfaces by randomlychoosing the values defining SE from an uniform distribution in the interval [−1, 1], where n is the totalnumber of integrations performed. For the multiple optimisation robustness analysis, the same set of n SEsurfaces are used to generate n optimised trajectories accounting for uncertainty.

III. System Models

The following section presents the mathematical models used to simulate the vehicle performance, specif-ically the propulsion system, vehicle aerodynamics, atmospheric model and the trajectory dynamics. As thevehicle configuration is based on a horizontal take-off, the full ascent profile should include a runway take-offfollowed by a subsonic climb-out. For this analysis however the complexities of the near-ground segmentwere ignored. The ascent profile instead starts just after the transition into the supersonic regime, at analtitude of around 8 km.

A. Dynamic model

The vehicle is considered to be a point with variable mass flying around a spherical, rotating earth. Thedynamics of the vehicle along the trajectory is governed by the following set of differential equations:16

h = v sin γ (2a)

v =FT cos ε−D

m− g sin γ + ω2

e(Re + h) cosλ (sin γ cosλ− cos γ sinχ sinλ) (2b)

γ =FT sin ε+ L

mvcosµ−

(g

v− v

Re + h

)cos γ + 2ωe cosχ cosλ (2c)

+ ω2e

(Re + h

v

)cosλ (sinχ sin γ sinλ+ cos γ cosλ)

χ =L

mv cos γsinµ−

(v

Re + h

)cos γ cosχ tanλ (2d)

+ 2ωe (sinχ cosλ tan γ − sinλ)− ω2e

(Re + h

v cos γ

)cosλ sin γ cosχ

λ =v cos γ sinχ

Rt + h(2e)

θ =v cos γ cosχ

(Rt + h) cosλ(2f)

6 of 18

American Institute of Aeronautics and Astronautics

where h is the altitude above mean sea level, v is the absolute velocity in a rotating Earth-centered referenceframe, γ is the flight path angle, χ is the path direction angle, µ is the bank angle, λ is the latitude, θis the longitude, m is the mass of the vehicle, FT is the magnitude of the thrust given by the engine, Land D are the aerodynamic lift and drag forces, respectively, Re = 6375 km is the mean Earth radius,ωe = 7.2921× 10−5 rad/s is the rotational velocity of the Earth, g0 = 9.80665 m/s2 is the acceleration dueto gravity at sea level and lastly ε = 0.08727 +α rad is the angle between the thrust vector and the velocityvector of the vehicle, accounting for a 5◦ thrust vector offset.

For simplicity, the control law only governs the angle of attack and thrust level, so no out of plane motionis considered (χ, µ = 0). In addition, the launch, ascent and orbit are all assumed to be contained withinthe equatorial plane (λ = 0).

B. Earth model

The gravitational field is assumed to be a function of the altitude and varies according to an inverse squarelaw g(h) = g0 (h/(Re + h))

2. The atmospheric characteristics (temperature, pressure, density and speed of

sound) follow the US Standard Atmosphere 1976 model up to 1000 km.

C. Propulsion model

As the vehicle is assumed to be propelled by a hybrid engine, two different models, one for the engine’sair-breathing mode and another for its rocket mode, were developed. For both modes, the models give themaximum available thrust FT and propellant consumption mP as a function of altitude h, Mach number Mand atmospheric conditions.

The numerical model for the air-breathing propulsion system is based on a modified analysis of a ramjet-type engine. The stagnation temperature T0 at the intake of the engine is a function of the flight Machnumber and is assumed to be that which results from isentropic compression of the oncoming flow. Lossesin the intake are manifest as a reduction in stagnation pressure at the exit of the intake diffuser compared tothe ideal case, as specified using a simple pressure recovery factor. For present purposes, a constant efficiencyis assumed for the intake, and this results in a pressure recovery factor between 0.75 and 0.95, depending onthe flight Mach number.

As the flight Mach number increases, the temperature rise due to compression in the intake increasesrapidly. If this temperature rise becomes too high then this limits the amount of energy that can be addedduring the combustion process and thus the thrust that can be produced by the engine. This effect ismitigated in the present analysis by setting the post-combustion temperature based on the flight Machnumber and altitude in order to ensure that sufficient energy can be input into the combustion chamber andthus an appropriate level of thrust can be maintained.

Fuel usage in the combustion chamber is determined from an energy balance across the combustionchamber, with the actual fuel-air ratio, fa, given by,

fa =cPgT04 − cPT03

ηcomb (Qr − cPgT04)(3)

where T03 and T04 are the temperatures at the start and end of combustion respectively, cP and cPg arethe specific heats of the gas before and after combustion respectively, Qr is the heating value of the fuel,here assumed to be hydrogen, and ηcomb is the combustion efficiency. A pressure drop across the combustionchamber of 4% is assumed.

The exit nozzle of the engine is assumed to expand the flow to ambient pressure, and the resulting nozzleexit static temperature is used to determine the exit Mach number and hence jet velocity. Expansion throughthe nozzle is assumed to have a constant adiabatic efficiency of 93%.

The overall mass flow for the engine, meng, is given by multiplying the nozzle exit area by the exit velocityand the static density. The fuel mass flow mPi for the engine is then obtained by multiplying the fuel-airratio, fa in Eq. (3), by the engine mass flow. The total fuel usage mP,ramjet of the vehicle is determinedby scaling mPi by the number of engines on the airframe, assuming all engines are operating at identicalconditions.

The total thrust for a single engine is given by the general thrust equation

FT,ramjet = meng ((1 + fa) ve − vfl) (4)

7 of 18

American Institute of Aeronautics and Astronautics

where ve is the nozzle exit velocity, and vfl is the flight velocity, determined from the flight Mach numberand atmospheric temperature. The total thrust delivered to the airframe is determined by scaling FT by thenumber of engines and again assuming that all engines are operating at the same conditions.

Figure 2 shows the overall thrust and fuel consumption as a function of altitude and Mach number forthe air-breathing engine model.

1015

2025

30

12

34

560

1

2

3

4

Altitude (km)Mach number

Thr

ust (

kN)

(a) Thrust FT,ramjet versus Mach number M andaltitude h

1015

2025

30

0

2

4

60

20

40

60

80

100

Altitude (km)Mach number

Fue

l mas

s flo

w (

kg/s

)

(b) Fuel consumption mP,ramjet versus Mach num-ber M and altitude h

Figure 2: Propulsion model outputs for the air-breathing ramjet-type engine.

Once the vehicle nears hypersonic speeds, the engine transitions from air-breathing mode to rocket mode.When the engine is in rocket mode, the basic rocket equation is used to determine the thrust level FT , andthe specific impulse Isp is then used to determine the propellant mass flow according to,

FT,rocket = mP ve + ∆PAe (5)

mP,rocket =Frocketg0Isp

(6)

where Ae is nozzle area at the exit, and ∆P is the difference between the free stream and exit pressures. Aspecific impulse of 460 s was assumed.

Finally, a throttle setting is applied as part of the control law such that both the applied thrust and massflow are scaled proportionally,

Fapplied = δTFT (7a)

mP,applied = δT mP (7b)

D. Aerodynamic model

A simple algebraic model is used to calculate the aerodynamic forces acting on the vehicle as a function ofangle of attack α and flight Mach number M . At low supersonic Mach numbers, the lift on the vehicle ismodeled as that due to a delta wing at a given angle of attack, as per linearized aerodynamic theory so that,

CL,ss =CLα√M2 − 1

Awing

Arefsinα cosα (8)

where Awing is the area of the lifting surfaces of the vehicle and CLα is a constant that is dependent on thegeometry of the vehicle. As the flight Mach number is increased towards the hypersonic regime, the non-linear variation of lift coefficient with angle of attack becomes progressively more important. An additionalcontribution to the lift coefficient,

CL,hs = 2Awing

Arefsin2 α cosα (9)

8 of 18

American Institute of Aeronautics and Astronautics

is thus calculated according to Newtonian flow theory17 and the variation with Mach number of the overalllift coefficient that is generated by the vehicle is then modeled as:

CL =CL,ss + CL,hs

2+√

(CL,ss)2 + (CL,hs)2 (10)

The drag coefficient of the vehicle is calculated simply as:

CD = CD0(M) + CL tanα (11)

where the first term CD0, a function of the Mach number, accounts for the wave, base, and viscous drag of

the vehicle and the second term is the induced drag of its lifting surfaces. The overall lift and drag forces onthe vehicle are then obtained by multiplying by the dynamic pressure q and the vehicle reference area Aref.

IV. Test Case

Due to the nature of the hybrid engine, the ascent trajectory is modeled as two phases: a first phase withthe engine in air-breathing mode, and a second phase with the engine in rocket mode.

For the trajectory optimisation problem, the control vector c = [ααα, δδδT , t1, t2], where ααα(t), δδδT (t) give thetime schedule of the angle of attack and throttle setting respectively, and t1, t2 are the durations of the firstand second phase with the engine mode switching from air-breathing to rocket propulsion at t1. The searchspace D for the controls is defined by the following bounds: α ∈ [−1, 1] rad, δT ∈ [0, 1] (i.e., from 0 to 100%of the maximum available thrust), and the flight time for each phase t1, t2 cannot exceed 1800 s, where thetotal flight time is tf = t1 + t2. The control law is the result of the optimal control problem maximizing thefinal vehicle mass,

maxc∈D

(m(t = tf )) (12)

subject to dynamics in Eq. (2) plus initial conditions set to start after the transition into the supersonicregime:

h(t = 0) = h0 = 8.2 km

v(t = 0) = v0 = 0.470 km/s

γ(t = 0) = γ0 = 8 deg

χ(t = 0) = χ0 = 0 deg

λ(t = 0) = λ0 = 0 deg

θ(t = 0) = θ0 = 0 deg

m(t = 0) = m0 = 261380 kg

(13)

The terminal conditions: h(t = tf ) = hf = 80 km, v(t = tf ) = vf = 7.380 km/s and γ(t = tf ) = γf =0 deg are those required to enter into a circular 80 km altitude orbit, in a rotating Earth reference frame.Link constraints are enforced between the two phases to guarantee the required continuity on both the statesand controls.

hf,1 = h0,2

vf,1 = v0,2

γf,1 = γ0,2

mf,1 = m0,2

αf,1 = α0,2

(14)

Additional path constraints based on operating conditions are imposed for both phases on the maximumaxial acceleration ax(t) ≤ 30 m/s2, maximum normal acceleration az(t) ≤ 30 m/s2, dynamic pressure throughthe equivalent air speed EAS(t) ≤ 335 m/s, and for the maximum Mach number in the air-breathing modeM(t ≤ t1) ≤ 5.4.

1. Shooting based evolutionary optimisation

For the evolutionary optimisation, the control vector c is composed of discrete values for the control law. Foreach solution, the control law is interpolated using a piecewise cubic interpolation and directly integrated

9 of 18

American Institute of Aeronautics and Astronautics

in time. The NLP problem to solve in this case is relatively small and can be handled by an evolutionaryapproach.

Within the control vector c, the angle of attack is discretized into 9 variables or elements for the firstair-breathing phase, c(1:9) ∈ [0, 30] deg and 8 elements for the second rocket phase, c(10:17) ∈ [−5, 30] deg.The throttle control setting for the first phase is preset to δT = 1, while the second phase is discretizedinto 9 elements c(18:26) ∈ [0.5, 1]. The last 2 design variables define the duration of first and second phaserespectively, c(27, 28) ∈ [80, 1800] s. The equality constraints on final states are converted into inequalityconstraints such that hf ∈ [80, 82] km, vf ∈ [7.34, 7.40] km/s, and γf ∈ [−2, 2] deg.

2. Finite element in time direct collocation

When solved by FET direct collocation method, the trajectory is decomposed into N elements, each of whichhas nc collocation points. After transcription, the optimal control problem including the initial and finalconditions given above becomes the general NLP problem with the objective function,

maxαααs,δδδs,ts

(m(t = tf )) (15)

subject to the nonlinear algebraic constraints in the form,

C(hs,vs,χχχs,λλλs, θθθs,ms,αααs, δδδs, ts) = 0 (16)

where hs, vs, χχχs, λλλs, θθθs, ms, αααs, δδδs, ts are vectors containing the set of discrete points for each of the state,control, and time variables at each node defined by the transcription scheme. Due to its high dimensionality ofthe NLP problem, a deterministic large-scale method was used over a more conventional stochastic approach,such DE.

A. Implementation of uncertainty

For the two atmospheric parameters, temperature and pressure, the uncertainty distribution function inEq. (1) is given by,

εatm(h) = lb,atm

(1− h

hc

)+ ub,atm

(h

hc

)(17)

where lb,atm and ub,atm are, respectively, the lower and upper boundaries defined as the percentage aroundthe nominal value, and h ∈ [0, hc] where hc = 150 km is the cut-off value for the increase, here set equal to themaximum value of the altitude vector. For the atmospheric temperature, the boundaries of the perturbationsare between lb,T = 10% and ub,T = 50%, while for pressure the bounds are set to between lb,P = 1% andub,P = 50%. Figure 3 shows an example of the approach applied to the temperature showing the nominalprofile compared to a perturbed profile due to uncertainties.

0 25 50 75 100 125 150100

200

300

400

500

600

700

800

900

1000

Altitude [km]

Tem

pera

ture

[K]

Perturbed profile splineNominal profileUpper boundLower bound

Figure 3: Nominal and perturbed values for the atmospheric temperature as a function of altitude.

In general, the level of uncertainty in atmospheric modeling increases with altitude. Specifically withthe US 1976 Standard Atmosphere model used here, below 32 km the model is well known and identical

10 of 18

American Institute of Aeronautics and Astronautics

to the Standard Atmosphere of the International Civil Aviation Organization (ICAO), however above that,and more notably above the ozone layer at 86 km the model becomes less accurate when compared withother complex models based on more recent experimental data.18 Other key contributors to uncertainty arethe stochastic nature of the radiation pattern from the Sun on the Earth’s surface and differences betweengeographic locations. A certain level of error is unavoidable when creating an averaged, globally applicableatmospheric model.

Aleatory uncertainties in the aerodynamic model arise, for instance, from the unknown effects of tur-bulence and any unsteadiness in the flow over the vehicle, while epistemic uncertainties, particularly in thehigh altitude or high speed regimes, arise from basic deficiencies in understanding the role of flow rarefactionand gas chemistry in governing the loads generated on the vehicle.

The uncertainty distribution function for the aerodynamic model is a function of three parameters withinthe model: the altitude, Mach number and angle of attack although is applied only to the lift and drag forcevariables. The bounds of the uncertainties are assumed to increase linearly with altitude, Mach number andthe absolute value of the angle of attack.

εaero(h,M,α) =

(lb,h

(1− h

hc

)+ ub,h

(h

hc

))+

(lb,M

(1− M

Mc

)+ ub,M

(M

Mc

))(18)

+

(lb,α

(1− α

αc

)+ ub,α

(α

αc

))where h ∈ [0, hc = 150] km, |α| ∈ [0, αc = 1] rad, M ∈ [0,Mc = 40]. The lower and upper bounds areset to 10% and 20% respectively, and are the same for all three parameters. As the uncertainty functionis dependant on multiple sources it means that the perturbations can be as much as ±30% of the nominalvalue when the vehicle is at Mach 0 at sea level with a 0◦ angle of attack, and can be as much as ±60%of the nominal value if the vehicle flies at Mach 40 at an altitude of 150 km with a 57.3◦ (1 rad) angle ofattack.

The uncertainty distribution function εaero is designed to reflect the expected margins of uncertainty ofthe aerodynamic model used here, which increases with the altitude and velocity since the model cannotaccurately predict the characteristics of the vehicle when flying at very high speed through the rarefied flowregime. The increase of the uncertainty bounds with the increase in angle of attack represents the inabilityto properly model the complex phenomena that occur at high angles of incidence, such as flow separation.

These uncertainty models, in particular the boundary conditions and uncertainty distribution functions,are general approximations and intentionally overestimated the potential uncertainties. Future work will beaimed at properly characterizing the uncertainties within the system by appropriate physical modelling.

V. Simulation Results

Following the procedures described above, a baseline trajectory was optimized using the nominal valuesfrom the models, i.e., not affected by any uncertainty. Figure 4 shows the optimized control law for the angleof attack α (see Fig. 4a) and the throttle setting δT (see Fig. 4b) as a function of time. The figures alsoshow the timings and transition point where the engine switches from air-breathing to rocket propulsion.

In order to analyze the sensitivity of the nominal control law to uncertainties in the system models,the set of controls shown in Fig. 4 were integrated 100 times with different randomized atmosphere andaerodynamic model parameters. The evolutions of state vectors with time are showed in Fig. 5. As canbe clearly seen none of the perturbed trajectories meet the final target conditions (h = 80 km, v = 7.38km/s, γ = 0◦). Figure 6 and Table 1 show the statistics on the deviation of the perturbed trajectories fromthe target conditions. In total, 94 of the 100 integrations converge towards the desired target conditionsfollowing a quasi-Gaussian distribution, while the remaining 6 computations diverge. Table 1 reports themean value and standard deviation of the final states of the system both when all runs are taken into account(‘full’) and when diverging runs are discarded from the set (‘clean’).

Similarly, Fig. 7 shows the constraints on the EAS and axial acceleration throughout the trajectory.Compared to the nominal trajectory, which meets exactly the limit of the constraint in both cases, themajority of the perturbed trajectories violate the maximum allowable EAS and ax values. Figure 8 andTable 2 show the statistics of the two constraint variables. In the case of the EAS, even with the outlyingvalues removed (so the clean set considering only 94/100 of the solutions), the mean EAS value is 431 m/scompared to the constraint limit of 335 m/s. The axial acceleration is slightly closer to the maximum limit

11 of 18

American Institute of Aeronautics and Astronautics

0 500 1000 1500 2000−10

−5

0

5

10

15

Time [s]

α [d

eg]

(a) Angle of attack

0 500 1000 1500 20000.5

0.6

0.7

0.8

0.9

1

1.1

Time [s]

Thr

ottle

(b) Throttle setting

Figure 4: Nominal control laws where the red lines refer to primary air-breathing phase, while the blacklines refer to the secondary rocket phase.

of 30 m/s2, with the full and clean sets having similar mean values of 30.16 m/s2 and 30.3 m/s2, respectively,with small standard deviations of 0.86 and 0.67.

Table 1: Statistics on the values of design parameters

Mean Standard deviation Mean Standard deviation

State (full) (full) (clean) (clean)

Altitude, h (m) 91649 51091 86158 29103

Velocity, v (m/s) 7114 1028 7370 250

Flight path angle, γ deg 6 29 -0.1 2.7

Final mass m, kg 56846 1941 56465 1293

Table 2: Statistics of path constraints.

Mean Standard deviation Mean Standard deviation

Constraint (full) (full) (clean) (clean)

Maximum EAS (m/s) 424 56.8 431 53

Maximum ax (m/s2) 30.16 0.86 30.3 0.67

Since the perturbed trajectories do not fully satisfy the various path and final constraints, the obtainedvalues for the final vehicle mass cannot be used as a basis for comparison. As such, a second approach wasused which performed 100 separate optimizations, each considering a different set of randomly perturbedatmospheric and aerodynamic parameters. The resultant control laws are shown in Fig. 9. The controllaws all follow the same pattern or shape as the nominal one, and are able to exactly meet the desired finalconditions (see Fig. 10) while also fulfilling all the path constraints. Figure 11 shows the values of the EASand axial acceleration for the entire trajectory, demonstrating that the two constraints are satisfied over thewhole trajectory.

The final vehicle masses that were obtained from the set of 100 optimizations of the control law canbe statistically analysed. The mean values of the final mass are quite similar (56752 kg for the multipleoptimisation case, and 56846 kg for the randomized integration of the nominal control law), while thestandard deviations are quite different, with 702 kg for the multiple optimisation case and 1941 kg for theperturbed integration of the nominal control laws.

12 of 18

American Institute of Aeronautics and Astronautics

0 500 1000 1500 20000

100

200

300

400

Time [s]

Alti

tude

[km

]

(a) Altitude

0 500 1000 1500 20000

2

4

6

8

Time [s]

Vel

ocity

[km

/s]

(b) Velocity

0 500 1000 1500 2000−20

−10

0

10

20

Time [s]

γ [d

eg]

(c) Path angle

0 500 1000 1500 2000

1

2

3x 105

Time [s]

Mas

s [k

g]

(d) Mass

Figure 5: Evolution of the states, where the bolded red and black lines refer to air-breathing and rocketphases respectively of the nominal trajectory, and the blue and green lines refer to air-breathing and rocketphases integrated with perturbations.

Figure 13 shows two cumulative distribution functions (CDF) representing the integration of the discretePDFs given in Fig. 12 and Fig. 6d. The information summarized in Fig. 13 can give an engineer, during thepreliminary design phase, an immediate understanding of the effect of uncertainties on the performance of thevehicle, shown here in terms of the final vehicle mass into orbit. The lower left side of the curve correspondsto the worst case scenario, when the effect of the uncertainty has the largest negative impact on the designwhich results in a very low final mass (in this analysis, every trajectory started with the same initial mass,thus the vehicle with largest final mass corresponds to the one with the lowest fuel consumption). On theother hand, the upper right side of the solid curve corresponds to the best case scenario, giving the highestmaximum final mass that can be obtained given the effects of the uncertainty.

As it is a maximisation problem that is being considered, the CDF curves give the complement of theexpectancy for obtaining a particular value of final mass between the best and worst case values. That is, ifthe uncertainties are correctly estimated, it can be guaranteed that in all cases (i.e., with a probability p = 1)that the vehicle will arrive in orbit with a final mass which is equal to the value obtained in the worst casescenario (54768 kg in the case of the solid green curve), while there is a probability p ≤ 0.01 that the vehiclewill arrive into orbit with a mass greater or equal to the value obtained in the best case scenario (58400 kgin this case). It is therefore possible to have a direct quantification of the probability of a given final massvalue between the best case and the worst case values, e.g., a final mass of 56188 kg can be obtained withp = 0.8, while a final mass of 57279 kg can be obtained only with p = 0.2.

It should be noted that while a sample size of only 100 is generally not considered large enough to drawaccurate conclusions from the statistics, the sample size is large enough to test the validity and usefulnessof this approach and to assess the potential for future work.

13 of 18

American Institute of Aeronautics and Astronautics

−100 0 100 200 300 400 5000

0.5

1

1.5

2x 10−5

Final Altitude [km]

PD

F

discreteparzengaussian

(a) Altitude

0 2 4 6 8 100

0.5

1

1.5

2x 10−3

Final Velocity [km/s]

PD

F

discreteparzengaussian

(b) Velocity

−20 −10 0 10 200

0.05

0.1

0.15

0.2

0.25

Final Path Angle [deg]

PD

F

discreteparzengaussian

(c) Path angle

5 5.5 6 6.5x 10

4

0

1

2

3

4

5

6x 10−4

Final Mass [kg]P

DF

discreteparzengaussian

(d) Mass

Figure 6: PDF of the final states when the nominal control laws are integrated by means of perturbedatmospheric and aerodynamic models.

VI. Conclusion

The paper describes a general approach for a trans-atmospheric trajectory optimisation. A hybridstochastic-deterministic algorithm and its application for the design of ascent trajectories for a hybrid-engine,high performance, unmanned, single-stage-to-orbit vehicle for payload deployment into low Earth orbit havebeen detailed. Uncertainties in atmospheric and aerodynamic models were considered, highlighting the needfor the effect of both epistemic and aleatory uncertainties to be considered from the very beginning of thedesign phase in order to gain an estimation of the impact on the performance of the vehicle. Integrating thetrajectory using a static, nominal optimal control law with randomly perturbed models allows for a sensi-tivity analysis of the obtained control law and the deviation of the results from the target final conditions tobe evaluated. A second multiple optimisation approach is then performed, giving a performance expectancycurve based on the objective function, e.g., the final mass, which can be used at the preliminary stage of thevehicle design to estimate the performance of the vehicle while accounting for uncertainties in the simulationmodels.

Future work will aim towards the implementation of further tools for the robust optimisation of trans-atmospheric trajectories, with the aim to optimise the expected values of performance while minimising theeffect of uncertainties. The authors are currently working on a more efficient clustering technique for themesh required by the FET method, and the identification of suitable methods and approaches which willbe able to solve large scale problems that are affected by noise and that are heavily constrained. Efficientsampling techniques are also being developed that, by reducing the noise, would allow the current hybridoptimisation tool to be applied to robust optimisation.

14 of 18

American Institute of Aeronautics and Astronautics

0 500 1000 1500 20000

500

1000

1500

EA

S [m

/s]

Time [s]

(a) Equivalent air speed

0 500 1000 1500 2000−20

−10

0

10

20

30

40

a x [m/s

2 ]

Time [s]

(b) Axial acceleration

Figure 7: Time evolution of the equivalent air speed (EAS) and axial acceleration ax, where the bolded redand black lines refer to air-breathing and rocket phases respectively of the nominal trajectory, and the blueand green lines refer to air-breathing and rocket phases integrated with perturbations.

0 200 400 600 8000

0.002

0.004

0.006

0.008

0.01

0.012

Max EAS [m/s]

PD

F

discreteparzengaussian

(a) Maximum equivalent air speed

25 30 350

0.2

0.4

0.6

0.8

1

Max Axial Acceleration [m/s2]

PD

F

discreteparzengaussian

(b) Maximum axial acceleration

Figure 8: PDF of constrained values when the nominal control laws are integrated by means of perturbedatmospheric and aerodynamic models.

References

1Varvill, R. and Bond, A., “The SKYLON Spaceplane,” Journal of the British Interplanetary Society, Vol. 57, 2004,pp. 22–32.

2Varvill, R. and Bond, A., “The SKYLON Spaceplane: Progess to Realisation,” Journal of the British InterplanetarySociety, Vol. 61, 2008, pp. 412–418.

3Butrica, A., Single Stage to Orbit: Politics, Space Technology, and the Quest for Reusable Rocketry, New Series in NASAHistory, The Johns Hopkins University Press, 2003.

4Betts, J. T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition,SIAM, 2010.

5Nocedal, J. and Wright, S., Numerical Optimization, Springer-Verlag GmbH, 2nd ed., 2006.6Costa, M. and Minisci, E., “MOPED: a Multi-Objective Parzen-based Estimation of Distribution algorithm,” Evolution-

ary Multi-Criterion Optimization. Second International Conference, EMO 2003 , edited by C. Fonseca, P. Fleming, E. Zitzler,K. Deb, and L. Thiele, Vol. 2632 of LNCS , Springer, Faro, Portugal, 08-11 April 2003, pp. 282–294.

7Vasile, M., Minisci, E., and Locatelli, M., “An Inflationary Differential Evolution Algorithm for Space Trajectory Opti-mization,” IEEE Transactions on Evolutionary Computation, Vol. 15, No. 2, 2011, pp. 267–281.

8Minisci, E., Campobasso, M. S., and Vasile, M., “Robust Aerodynamic Design of Variable Speed Wind Turbine Rotors,”Accepted for ASME Turbo Expo 2012 , Copenhagen, Denmark, June 11-15 2012.

15 of 18

American Institute of Aeronautics and Astronautics

0 500 1000 1500 2000−10

−5

0

5

10

15

Time [s]

α [d

eg]

(a) Angle of attack

0 500 1000 1500 20000.5

0.6

0.7

0.8

0.9

1

1.1

Time [s]

Thr

ottle

(b) Throttle setting

Figure 9: Optimized control laws where the bolded red and black lines refer to air-breathing and rocketphases respectively of the nominal trajectory, and the blue and green lines refer to air-breathing and rocketphases integrated with perturbations.

9Lozano, J. A., Larranaga, P., and Inza, I., Towards a New Evolutionary Computation: Advances on Estimation ofDistribution Algorithms (Studies in Fuzziness and Soft Computing), Springer, February 2006.

10Fukunaga, K., Introduction to statistical pattern recognition, Academic Press, 1972.11Avanzini, G., Biamonti, D., and Minisci, E., “Minimum-Fuel/Minimum-Time Maneuvers of Formation Flying Satellites,”

Astrodynamics Specialist Conference, AAS 03-654, Big Sky, Montana, 03-07 August 2003.12Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T., “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II,”

IEEE Transaction on Evolutionary Computation, Vol. 6, No. 2, april 2002, pp. 182–197.13Price, K., Storn, R., and Lampinen, J., Differential Evolution: A Practical Approach to Global Optimization, Springer,

2005.14Leary, R., “Global optimization on funneling landscapes,” Journal of Global Optimization, Vol. 18, No. 4, 2000, pp. 367–

383.15Vasile, M., “Finite elements in time: a direct transcription method for optimal control problems,” AIAA/AAS Astrody-

namics Specialist Conference, AIAA Paper 2010-8275, Toronto, Canada, August 2-5 2010.16Miele, A., Optimal trajectories of aircraft and spacecraft , No. 19910001655, JPL-956415; NAG1-516 in AGARD, Aircraft

Trajectories: Computation, Prediction, Control, NASA, 1990.17Anderson, J. D., editor, Fundamentals of Aerodynamics, McGraw-Hill Higher Education, 3rd ed., 2001.18AIAA Guide to Reference and Standard Atmosphere Models (G-003C-2010e), 2010.

16 of 18

American Institute of Aeronautics and Astronautics

0 500 1000 1500 20000

20

40

60

80

100

Time [s]

Alti

tude

[km

]

(a) Altitude

0 500 1000 1500 20000

2

4

6

8

10

Time [s]

Vel

ocity

[km

/s]

(b) Velocity

0 500 1000 1500 2000−5

0

5

10

15

Time [s]

γ [d

eg]

(c) Flight path angle

0 500 1000 1500 20000.5

1

1.5

2

2.5

3x 105

Time [s]

Mas

s [k

g]

(d) Mass

Figure 10: Evolution of the states for optimized control law where the bolded red and black lines refer toair-breathing and rocket phases respectively of nominal trajectory, and the green and magenta lines refer tooptimised air-breathing and rocket phases with perturbations.

0 500 1000 1500 20000

50

100

150

200

250

300

350

EA

S [m

/s]

Time [s]

(a) Equivalent air speed

0 500 1000 1500 2000−10

0

10

20

30

40

a x [m/s

2 ]

Time [s]

(b) Axial acceleration

Figure 11: Evolution in time of equivalent air speed and axial acceleration, where the bolded red and blacklines refer to air-breathing and rocket phases respectively of nominal trajectory, and the green and magentalines refer to optimised air-breathing and rocket phases with perturbations.

17 of 18

American Institute of Aeronautics and Astronautics

5.3 5.4 5.5 5.6 5.7 5.8 5.9 6x 10

4

0

0.2

0.4

0.6

0.8

1x 10−3

Final Mass [kg]

PD

F

discreteparzengaussian

Figure 12: PDF of final mass as obtained by op-timized trajectories.

5.2 5.4 5.6 5.8 6 6.2 6.4x 10

4

0

0.2

0.4

0.6

0.8

1

Final Mass [kg]

Pro

babi

lity

p

Figure 13: Comparison of CDF curves of finalmasses for the two approaches where the greensolid curve shows the optimized trajectories andred dashed curve, those integrated with pertur-bations.

18 of 18

American Institute of Aeronautics and Astronautics